^^^ Tommaso is the Italian version of Thomas as well.
Like I mentioned above, when Italians call Gaetano, they usually say, Tano. I think this has mutated into Tom or Tommy. It has a lot to do with dialect, not always making grammatical sense.

Take for instance Calogero and how that becomes Luca (and this is italian to italian lol). Italian nickname for Calogero (which is generally a sicilian name) in which Sicilians will call, Caluzzu, which will become Luzzu, which becomes Luca. It doesn't have to make sense but things happen within different dialects.